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# Maria Infusino

INdAM-COFUND Outgoing fellow between 2012-11-05 and 2014-11-04

### Project information

#### Title

RQRP: Realizability and Quantum Representability Problem

#### Outgoing host organisation Return host organisation

University of Reading Università degli Studi di Roma “La Sapienza”

Whiteknigths House Piazzale Aldo Moro, 5

PO BOX 217 00185 Roma, Italy

Reading RG66AH

United Kingdom

http://www.reading.ac.uk/ http://www.uniroma1.it/

#### Abstract

Complex systems, such as many-body systems like liquid composed of molecules, are generally very difficult to describe. The investigation of such systems is greatly facilitated if the attention is restricted to selected physical parameters (usually correlation functions) which encode the relevant structure of the system. The realizability and the quantum representability problems exactly address the question of identifying conditions to determine whether a putative characteristic can be actually realized by a state of the underlying system.

Realizability and representability arise repeatedly in different areas, thus they are promising viewpoints on complex systems. Recently, there has been a revival of interest in these problems in statistical mechanics as well as in quantum\ chemistry but progress is hindered by a lack of understanding of their mathematical structure.

The primary aim of this project is to establish rigorous mathematical foundations for the analysis of large complex systems. The innovative strategy proposed is to tackle the realizability problem from the point of view of moment theory. In fact, the finite dimensional projections of the realizability problem give a range of truncated moment problems of increasing dimension. This new approach is thought to shed some light on the problem to identify further relevant explicit conditions for realizability out of the rather inexplicit ones known in literature.

The representability problem is essentially a quantum version of the realizability problem. The methods developed in this project for the realizability problem can be, therefore, used to derive an explicit solution of the quantum representability for low densities.

### Fellow information

#### Research interests

Measure theory, probability, number theory,

mathematical physics and functional analysis.

In particular:

- Realizability of correlation functions and quantum representability problem.
- Truncated moment problem and its applications.
- Uniform distribution theory.
- Random sequences of Steiner symmetrizations.

#### Previous positions, awards

*2011-2012*: Postdoctoral Research fellow, Department

of Mathematics, University of Reading, UK

Research project:*Truncated*

moment problems in Statistical Physics and Quantum Chemistry.

Principal investigator: Dr Tobias Kuna.*February 2011*: PhD in Mathematics and Computer Science

at University of Calabria, Italy.

Thesis title:*Uniform*

distribution of sequences of points and partitions.

Supervisor: Prof. Aljosa Volcic.*October 2007*: MSc in Mathematics*cum Laude*at University of Calabria, Italy.

Thesis title:*Uniformly distributed sequences on fractal sets*

Supervisor: Prof. Aljosa Volcic.*October 2005*: BSc in Mathematics*cum Laude*at University of Calabria, Italy.

Thesis title:*Newton-Kantorovich method for non-lin ear operators in*

Banach space

Supervisor: Prof. Espedito De Pascale.*July 2012*: Prize for*The best research poster*at

the 6th European Congress of Mathematics (6ECM), Jagiellonian

University, KrakÃ³w, Poland.*July 2008*: Awarded*The best graduate in*from the School of Mathematics, Physics and Natural Sciences

2007

of University of Calabria, Italy.*2002-2007*: Fellowship from*Fondazione*. (Academic regional association which

Calabria Scienze Oggi

supports outstanding students, providing them with a more advanced

quality education and training.)

#### Publications, preprints, other works

- M. Infusino and A. Volcic,
*Uniform distribution on fractals*, Uniform Distribution Theory, 4 (no.2): 47-58, 2009 - M. Drmota and M. Infusino,
*On the discrepancy of some generalized Kakutani’s sequences of partitions*, Uniform Distribution Theory, 7 (no.1): 75â104, 2012 - M. Infusino, T. Kuna, and A. Rota,
*Concrete conditions for realizability of moment measures via quadratic modules*, in preparation - M. Infusino, T. Kuna, J. L. Lebowitz, and E. R. Speer,
*Truncated moment problem as a special case of realizability*, in preparation

#### Conferences, schools, other events

*Polynomial Optimisation: Summer School and Workshop*

Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 15-19 July 2013*Stochastic and Infinite Dimensional Analysis*

ZIF, University of Bielefeld, Germany, 24-28 June 2013*Young women in discrete mathematics*

Bonn, Germany, 7-9 June 2013*Research visit*

Rutgers University, USA, 26 March – 26 April 2013*Analysis day*

University of Reading, UK, 19 October 2012*6th European Congress of Mathematics*

Jagiellonian University, Poland, 1-8 July 2012*107th Statistical Mechanics Conference*

Rutgers University, USA, 8 May 2012*XIX Conference of Italian Mathematical Society (UMI)*

University of Bologna, Italy, 12-17 September 2011*Spring School in Discrete Probability, Ergodic Theory and Combinatorics*

TU-Graz, Austria, 4-15 April 2011*II International Conference on Uniform Distribution Theory*

Strobl, Austria, 5-9 July 2010